Sign patterns of rational matrices with large rank II

It is known that, for any real m-by-n matrix A of rank n-2, there is a rational m-by-n matrix which has rank n-2 and sign pattern equal to that of A. We prove a more general result conjectured in the recent literature.

Disentangling environmental effects in microbial association networks

Background Ecological interactions among microorganisms are fundamental for ecosystem function, yet they are mostly unknown or poorly understood. High-throughput-omics can indicate microbial interactions through associations across time and space, which can be represented as association networks. Associations could result from either ecological interactions between microorganisms, or from environmental selection, where the association is environmentally driven. Therefore, before downstream analysis and interpretation, we need to distinguish the nature of the association, particularly if it is due to environmental selection or not. Results We present EnDED (environmentally driven edge detection), an implementation of four approaches as well as their combination to predict which links between microorganisms in an association network are environmentally driven. The four approaches are sign pattern, overlap, interaction information, and data processing inequality. We tested EnDED on networks from simulated data of 50 microorganisms. The networks contained on average 50 nodes and 1087 edges, of which 60 were true interactions but 1026 false associations (i.e., environmentally driven or due to chance). Applying each method individually, we detected a moderate to high number of environmentally driven edges—87% sign pattern and overlap, 67% interaction information, and 44% data processing inequality. Combining these methods in an intersection approach resulted in retaining more interactions, both true and false (32% of environmentally driven associations). After validation with the simulated datasets, we applied EnDED on a marine microbial network inferred from 10 years of monthly observations of microbial-plankton abundance. The intersection combination predicted that 8.3% of the associations were environmentally driven, while individual methods predicted 24.8% (data processing inequality), 25.7% (interaction information), and up to 84.6% (sign pattern as well as overlap). The fraction of environmentally driven edges among negative microbial associations in the real network increased rapidly with the number of environmental factors. Conclusions To reach accurate hypotheses about ecological interactions, it is important to determine, quantify, and remove environmentally driven associations in marine microbial association networks. For that, EnDED offers up to four individual methods as well as their combination. However, especially for the intersection combination, we suggest using EnDED with other strategies to reduce the number of false associations and consequently the number of potential interaction hypotheses.

On the infinite Borwein product raised to a positive real power

In this paper, we study properties of the coefficients appearing in the q-series expansion of $$\prod _{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta $$ ∏ n ≥ 1 [ ( 1 - q n ) / ( 1 - q pn ) ] δ , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power $$\delta $$ δ . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For $$p=3$$ p = 3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $$\delta $$ δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $$p=3$$ p = 3 case.

On the Eventual Exponential Positivity of Some Tree Sign Patterns

An n×n matrix A is called eventually exponentially positive (EEP) if etA=∑k=0∞tkAkk!>0 for all t≥t0, where t0≥0. A matrix whose entries belong to the set {+,−,0} is called a sign pattern. An n×n sign pattern A is called potentially eventually exponentially positive (PEEP) if there exists some real matrix realization A of A that is EEP. Characterizing the PEEP sign patterns is a longstanding open problem. In this article, A is called minimally potentially eventually exponentially positive (MPEEP), if A is PEEP and no proper subpattern of A is PEEP. Some preliminary results about MPEEP sign patterns and PEEP sign patterns are established. All MPEEP sign patterns of orders n≤3 are identified. For the n×n tridiagonal sign patterns Tn, we show that there exists exactly one MPEEP tridiagonal sign pattern Tno. Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of Tno. We also classify all PEEP star sign patterns Sn and double star sign patterns DS(n,m) by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively.